All Questions
6,260 questions
-1
votes
2
answers
806
views
The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
- 251
25
votes
3
answers
4k
views
Largest number of vectors with pairwise negative dot product
What is the largest $m$ such that there exist $v_1,\dots,v_m \in \mathbb{R}^n$ such that for all $i$ and $j$, $1\leq i< j\leq m$, we have $v_i \cdot v_j < 0$.
Also, the preview screen is not ...
user5810
2
votes
0
answers
1k
views
Good sources for linear algebra for convex optimization and graph analysis?
What are some good sources for linear algebra for convex optimization and graph analysis?
In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses (...
- 2,383
28
votes
5
answers
4k
views
Does Smith normal form imply PID?
Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we ...
user5810
12
votes
3
answers
3k
views
How to combine linear constraints on a matrix and its inverse?
Suppose there exists a $(n \times n)$ matrix $A$ that is real and invertible (nothing unusual or special about $A$). We do not know the entries of $A$. However, we do have linear constraints, some of ...
6
votes
5
answers
1k
views
A signature inequality?
Given two real symmetric matrices $A$ and $B$ of common square size $n$ with no strictly negative eigenvalues, can the symmetric matrix $AB+BA$ have strictly more than $n/2$ eigenvalues which are ...
- 17.9k
6
votes
2
answers
2k
views
Inner products and norms
Let $f:[n]\times [n] \rightarrow [0,1]$ be a function from pair of integers to the real interval [0.1]. I would like to find sets of complex vectors
$X= \{x_i\}$ and $Y=\{y_j\}$ satisfying $x_i\cdot ...
1
vote
2
answers
1k
views
Assessing measurement accuracy and precision
I have been asked to assess the accuracy and precision of a new measurement method (Let's call it method B). This new method is compared to an existing one (A) that has its own specifications in ...
- 31
4
votes
1
answer
496
views
Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 711
0
votes
2
answers
372
views
Quantum observables
Let H be a Hilbert space and A, B two non-commuting bounded linear operators. Let Com(A,B) be the set of bounded linear operators C which commute both with A and B.
Question 1 : What is known about ...
8
votes
3
answers
570
views
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
- 5,422
10
votes
3
answers
6k
views
Solving a system of linear inequalities -- what is the dimension of the solution set?
It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$?
For the applications I have in mind the ...
- 7,857
12
votes
2
answers
9k
views
Is there a way to simplify block Cholesky decomposition if you already have decomposed the submatrices along the leading diagonal?
Let's say we have a block matrix $ M =\left( \begin{array}{ccc}
A & B\\
B^{*} & C \end{array} \right)$ where $M$ is positive definite. ($A$ and $C$ are also positive definite.)
There is a ...
9
votes
1
answer
1k
views
(Elementary?) combinatorial identity expressing binomial coefficients as an alternating sum over permutations.
Background
I came up with this while trying to find a sort of high-level exposition of the exterior algebra of a vector space. Let $V$ be a vector space of dimension $n$ over $\mathbb{C}$, and let $...
- 8,559
6
votes
2
answers
2k
views
Efficient approximation of a matrix and its inverse
Assume that $ A $ is a real $ n\times n $ matrix whose rows constitute an orthonormal basis of $ \mathbb R^n $.
Informal statement of question: Assume we want to approximate $ A $ by a rational ...
- 201
6
votes
2
answers
8k
views
Existence/Uniqueness of Nonnegative Solutions of Linear Systems of Equations
Suppose we have an $m$x$n$ matrix $A$, with $m\lt n$, and an $m$x$1$ vector $b$. Are there existence and uniqueness conditions characterizing nonnegative solutions of the system of linear equations $...
3
votes
0
answers
2k
views
Eigenvalue Problems with Linear Constraints
The motivation for this problem comes from trying to develop a simple way to decompose domains into non-overlapping subdomains to solve for the eigenvalues of some differential operator. The idea is ...
- 543
2
votes
1
answer
510
views
hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
2
votes
0
answers
4k
views
Eigenvalues of sum of commuting matrices [closed]
With reference to the following thread :
Eigenvalues of Matrix Sums
Answer by Jonas Meyer is as follows :
If 2 positive matrices commute, than each eigenvalue of the sum is a sum of eigenvalues of ...
- 21
1
vote
1
answer
738
views
Matrix Conjugates over Finite Fields
Thinking about Diffe-Hillman for matrices brought me to the following question.
Given $\mathbb{F}_{p^k}$ the finite field with $p^k$ elements when can we find non-trivial solutions to
$\begin{...
- 4,842
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 139
61
votes
11
answers
11k
views
Geometric proof of the Vandermonde determinant?
The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
- 23k
3
votes
2
answers
376
views
Invariant subspaces of subalgebras of $M_n(C)$
Given a subalgebra E of $M_n$ (nxn complex valued matrices), what can we say about the subspaces F of $M_n$ such that $EF \subset F$? Googling for an answer gives me the reference:
Israel Gohberg, ...
- 243
4
votes
1
answer
548
views
O(n^2) algorithm to approximate the sum of the log of the singular values of a matrix
Given an $M \times N$ matrix of rank $N$ ($M \ge N$) with $i^{th}$ singular value $\sigma_i$, does their exist an $O(M^2)$ algorithm for approximating the sum $ H =\sum_{i=1}^N \log(\sigma_i)$ with ...
- 188
7
votes
5
answers
2k
views
Explicit invariants (under change of basis) of maps $V \to V \otimes V$.
It is standard to construct numbers associated to a linear transformation $f: V \to V$ of a finite-dimensional vector space which are invariant under change of basis. The coefficients of the ...
- 5,010
1
vote
1
answer
715
views
Find a matrix's nullspace from submatrix nullspace
This is probably a basic question, but my linear algebra is weak.
Suppose I want to compute the nullspace of a matrix A using some iterative method (e.g. Lanczos). Suppose further that I know a ...
9
votes
2
answers
2k
views
Jordan Form Over a Polynomial Ring
Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (...
- 2,097
2
votes
1
answer
810
views
On matrices that almost have the same eigenvalues
Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if
$$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and ...
- 2,154
3
votes
1
answer
633
views
What is the entropy of a density matrix which is the sum of two unitarily equivalent projectors?
Construction
Suppose I have a density matrix $\rho$ which is proportional to a projector $P$ formed by tensoring together $N$ small projectors $P^{(i)}$ of rank 2:
$P^{(i)} = |a\rangle_i\langle a| + |...
- 846
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
- 27.9k
1
vote
2
answers
917
views
Any known compact expression for
Is there any known compact expression for the sum
$$S_{k} = \sum_{i=1}^{k} A^{i-1} P Q^{k-i}$$
where $A$, $P$ and $Q$ are respectively $m \times m$, $m \times n$ and $n \times n$ matrices?.
You can ...
- 61
4
votes
1
answer
866
views
When is a triangular matrix totally unimodular?
I have a {0,1}, invertible, triangular matrix, that I would like to show is totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?
- 1,182
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
22
votes
3
answers
3k
views
Splitting the determinant polynomial into linear factors - a Dedekind problem
Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial
$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\...
- 33.8k
-4
votes
2
answers
6k
views
Factorizing polynomials of several variables (in a different perespective)
I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^...
1
vote
1
answer
357
views
Finding the $J$ for a symplectic vector space
I found something strange when I was working on some other problems.
I thought the triple intersection description of the unitary group said that any two of $(g, \omega, J)$ determines the third ...
- 1,525
11
votes
1
answer
688
views
Existence of a pair of matrices in SL(2,Z) satisfying certain constraints on the spectral radius
Some background: my coauthors and I are working on a problem which deals with the exponential growth rates of certain infinite products of matrices. One of the sub-problems which arises in this ...
- 6,206
2
votes
0
answers
5k
views
A system of linear equations with linear constraints
Mathematical problem.
Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of $\...
14
votes
2
answers
937
views
Involutions in GL_n(Z)
Is there a classification of involutions in $\text{GL}_n(\mathbb{Z})$?
Here's some more details about what I mean. Consider $f \in \text{GL}_n(\mathbb{Z})$ such that $f^2=1$. Regard $f$ as an ...
- 245
8
votes
2
answers
3k
views
Centralizers in GL(n,p)
There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is ...
- 10.7k
4
votes
0
answers
790
views
Is it possible to use linear programming to solve this problem?
I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group.
Could someone comment on whether this is ...
- 41
13
votes
4
answers
2k
views
Rational congruence of binomial coefficient matrices
Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on.
Say that two matrices $A$ and $B$ over the ...
- 82.7k
4
votes
2
answers
1k
views
Simultaneous Block decomposition of a set of orthogonal projections
An orthogonal projection is an Hermitian matrix $P$ such that $P^2=P$.
Denote $U^*$ the conjugate transpose of a matrix $U$.
It can be easily shown that for two projections $P_1$ and $P_2$, there ...
9
votes
3
answers
4k
views
Eigenvectors of a certain big upper triangular matrix
I'm looking at this matrix:
$$
\begin{pmatrix}
1 & 1/2 & 1/8 & 1/48 & 1/384 & \dots \\
0 & 1/2 & 1/4 & 1/16 & 1/96 & \dots \\
0 & 0 & 1/8 & 1/16 &...
2
votes
1
answer
456
views
A question on matrix decomposition.
Is the following claim true?
Claim Let $A, B\in C^{n\times n}$ with $rank(A)=rank(B)=r$. Then there exist nonsingular matrices $P_1, P_2, Q_1, Q_2$ such that
$$ Q_1AP_1=Q_2BP_2=\left(\begin{array}{...
- 1,858
0
votes
2
answers
580
views
Linear algebra inequality
I'm wondering (hoping) if an inequality is true. Please can anyone help me?
Let $V$ be a complex vector space $dim_{\mathbb{C}}(V)=n$
with a hermitian scalar product $h$.
Let $v,a, b \in V$.
Is it ...
- 1,727
12
votes
0
answers
349
views
Matroids with prescribed independent sets
Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...
- 1,791
15
votes
9
answers
9k
views
Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
- 151
4
votes
2
answers
16k
views
Submultiplicative matrix norm: Max Norm
Various sources claim that a maximum norm $||A||_{max}=\max_{i,j}|a_{ij}|$ is not submultiplicative, i.e. $||AB||_{max}\not\leq||A||_{max}||B||_{max}$.
Where can I find what norm a,b satisfy $||AB||...
- 93
3
votes
2
answers
2k
views
How can we explicitly find the maximum eigenvalue of a tridiagonal matrix?
I just came across a matrix of the form
$A:=\begin{pmatrix}
0&-\frac{c_0}{b_0}&0&\cdots&0\\-\frac{a_1}{b_1}&0&-\frac{c_1}{b_1}&\cdots&0\\0&-\frac{a_2}{b_2}&0&...
- 93